I''m trying to do a space shooter (like Subspace). When the ship shoots, should the speed of the ''bullet'' be a constant or do I have to add the spaceship''s speed to the bullet too?

Two points to consider:

If the bullets speed is substantially more than the ships ( say 100x greater ), than no.

else if the bullets speed is not much greater than the ships, than you need to, but then the bullets are more like some type of torpedo or missle.
If this is true, you need to be carefull of the scale of the speeds, unless you want to have to consider relativistic effects ( which you don''t want to do ).


Best case: Have the bullets go much faster than the ships, but much slower than the speed of light ( unless it is a light weapon ).